The present invention relates to apparatuses and methods for making interleaved cylindrical rolls from a polymer roll and a porous roll. The interleaved rolls disclosed herein are useful for subsequent high pressure gas impregnation and solid-state foam processing. In one embodiment, an interleaving apparatus comprises: a first rotatable platform for unwinding a first cylindrical roll of a first sheet material, the first rotatable platform being rotatable about a first central axis; a second rotatable platform for unwinding a second cylindrical roll of a second sheet material, the second rotatable platform being rotatable about a second central axis; and a third rotatable platform for winding, while the first and second materials are unwinding, the first sheet material together with the second sheet material to yield the interleaved cylindrical roll, the third rotatable platform being rotatable about a third axis and proximate to the first and second rotatable platforms.